We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy. Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces $H_D^{-\zeta,p}$ with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator.

F. Tröltzsch, Optimal Control of Partial Differential Equations, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.
doi: 10.1090/gsm/112.

[65]

L. Weis, A new approach to maximal $L_p$-regularity, in Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), vol. 215 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2001, 195-214.

F. Tröltzsch, Optimal Control of Partial Differential Equations, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010, Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.
doi: 10.1090/gsm/112.

[65]

L. Weis, A new approach to maximal $L_p$-regularity, in Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), vol. 215 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 2001, 195-214.

Ugur G. Abdulla.
On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences.
Inverse Problems & Imaging,
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Ugur G. Abdulla.
On the optimal control of the free boundary problems for the
second order parabolic equations. I. Well-posedness and convergence of the method of lines.
Inverse Problems & Imaging,
2013, 7
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doi: 10.3934/ipi.2013.7.307